If you’ve ever taught a differential equations course and yearned for fresh examples, this is definitely a book for you. Wouldn’t it be great to give some of those terribly overworked examples a rest? Even some of the more recent examples from chemistry and biology are starting to look a bit fatigued.

This book was written by two earth scientists with the aim of providing what they regard as essential skills for graduate students and advanced undergraduates in their field. Those skills include the ability to translate chemical and physical systems into mathematical and computational models that provide insight into dynamical processes on the earth’s surface and inside. All the models treated here are based on ordinary and partial differential equations.

What kinds of examples do the authors use? One of the first is a model of the radiocarbon content of the biosphere. In simplest form, this leads to exponential decay; with periodic forcing (determined perhaps by the sunspot cycle), more complicated solutions arise. As the book proceeds, the examples get more complex. Other examples include: dissolved species in an aquifer, evolution of a sandy coastline, and pollutant transport in a confined aquifer. One amazing pictorial example from the introduction shows a simulated time sequence of an iron bolide asteroid, one kilometer in diameter, hitting the ocean at a 45 degree angle. Two of the most interesting worked out examples are analysis of complicated circulation patterns in Lake Ontario and modeling a lahar (water and pyroclastic debris flowing down the side of a volcano).

Although the authors concentrate on earth science, pretty much everything is accessible to anyone having some knowledge of basic physics and chemistry. The early chapters of the book pay a lot of attention to the details of developing a model. It is clear that the authors have a good deal of expertise in modeling. Yet they note that their book is intended as a primer, and students are not expected to have any modeling experience. By the nature of their interests, most of the discussion revolves around partial differential equations — nearly all the problems of interest have time and at least one spatial dimension as independent variables. One consequence of this is a secondary focus in the book on finite difference methods for solving partial differential equations.

Besides being a wonderful source of examples, this short book is a pleasing and well-organized introduction to modeling with differential equations.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.